Axiomatic Type Theory Overview

• Axiomatic Type Theory is defined as a dependent type theory that replaces judgmental computation rules with propositional computation axioms using identity types.
• It bridges proof theory, categorical semantics, and homotopical models by incorporating structures such as path categories and two-level type theory.
• The framework enables controlled addition of extensionality and preservation of metatheoretic properties like decidability and confluence.

Searching arXiv for recent and foundational papers on axiomatic type theory, two-level type theory, and semantics. Axiomatic type theory denotes a family of dependent type theories in which the usual computation rules are not imposed as judgemental equalities, but are instead replaced by computation axioms typed by identity types. In the formulation explicitly described for dependent type theory with computation axioms, the term equality judgements that usually characterise computation rules are replaced by additional term judgements expressing those equations propositionally (Spadetto, 9 Jul 2025). In a closely related 2025 account of axiomatic Martin-Löf type theories, axiomatic identity types are specified so that the usual definitional $\beta$-reduction rule is weakened to a propositional $\beta$-axiom (Otten et al., 19 Mar 2025). The topic therefore sits at the intersection of proof theory, categorical semantics, and homotopical approaches to type theory: it isolates intensional structure while weakening definitional computation, and it connects naturally to two-level type theory, path categories, display map 2-categories, and general semantic frameworks for type theories (Annenkov et al., 2017, Uemura, 2019).

1. Definition and basic syntactic idea

Axiomatic type theory is described in the 2025 2-categorical account as “a dependent type theory without computation rules” (Spadetto, 9 Jul 2025). In that setting, one keeps the usual formation, introduction, and elimination rules of intensional Martin-Löf type theory for each type former, but replaces each computation rule by a propositional equality. For identity types, the eliminator $J$ is retained, yet instead of the strict judgemental equation

$J(c,x,x,r(x)) \equiv c(x),$

one postulates only the axiom

$H(c,x): J(c,x,x,r(x)) = c(x),$

where “$=$” is the identity type of the target family (Spadetto, 9 Jul 2025).

The 2025 biequivalence result presents the same weakening in proof-theoretic terms. For each type judgement $\Gamma \vdash A\ \mathrm{type}$ and endpoints $a,a':A$, one has the identity type $\Gamma \vdash a =_A a'\ \mathrm{type}$, reflexivity, and a based eliminator, but the definitional $\beta$-rule is replaced by a propositional inhabitant

$\beta$0

of an identity type (Otten et al., 19 Mar 2025). That paper calls these “axiomatic Id-types,” and notes that such Id-types appear in cubical type theory as well as in “type theory without definitional equality” (Otten et al., 19 Mar 2025).

This syntactic shift changes the status of computation from conversion to proof. A plausible implication is that the boundary between syntax and semantics is redrawn: equations that are built into type checking in stricter systems become explicit inhabitants of identity types in axiomatic systems. The supplied materials make this point directly by contrasting judgemental equality with propositional equality in the interpretation of computation rules (Spadetto, 9 Jul 2025).

2. Identity types and computation axioms

Identity types are the central example through which axiomatic type theory is characterized. In the axiomatic setting described in 2025, one often uses the based Paulin-Möhring eliminator, fixing one endpoint and requiring only a propositional computation law (Otten et al., 19 Mar 2025). The strict intensional condition

$\beta$1

is explicitly not assumed; instead, one postulates an inhabitant of the appropriate identity type (Otten et al., 19 Mar 2025).

The same weakening appears in the broader ATT presentation. There, the usual identity-type rules are retained:

• formation from $\beta$2,
• reflexivity $\beta$3,
• elimination by $\beta$4,

but the computation principle is only

$\beta$5

as a term of an identity type (Spadetto, 9 Jul 2025).

This proof-relevant treatment of computation extends beyond identity types. The 2025 2-categorical account states that similar figures—formation, introduction, elimination, and computation axiom—arise for $\beta$6, $\beta$7, $\beta$8, $\beta$9, $J$0, $J$1, and function extensionality (Spadetto, 9 Jul 2025). Likewise, the path-category account describes weakly stable $J$2-types with $J$3- and $J$4-rules, and explains that after strictification one obtains a strict model of axiomatic Martin-Löf type theory (Otten et al., 19 Mar 2025).

A related but more specialized development is the axiomatization of typal heterogeneous equality. That work postulates a heterogeneous equality type

$J$5

together with eight primitive operations and typal, i.e. propositional, computation properties (Pitts, 2019). It derives a corrected coercion $J$6 whose regularity is itself propositional, and proves uniqueness of identity proofs and Streicher’s Axiom K from these axioms (Pitts, 2019). This is not presented as a general definition of ATT, but it shows that axiomatic presentations of equality can be both minimal and sufficient to recover standard eliminatory behavior up to propositional equality.

3. Two-level type theory as an axiomatic framework

Two-level type theory provides a distinct but closely connected manifestation of axiomatic type-theoretic methodology. It is defined as a version of Martin-Löf type theory combining two different type theories, called the inner and the outer theory (Annenkov et al., 2017). In the case emphasized there, the inner theory is homotopy type theory, possibly with univalent universes and higher inductive types, while the outer theory is an extensional theory validating uniqueness of identity proofs (Annenkov et al., 2017).

The detailed presentation distinguishes shared contexts and two layers of judgements:

• outer types $J$7 and terms $J$8,
• inner types $J$9 and terms $J(c,x,x,r(x)) \equiv c(x),$0, together with two equality formers:
• outer $J(c,x,x,r(x)) \equiv c(x),$1,
• inner paths $J(c,x,x,r(x)) \equiv c(x),$2 (Annenkov et al., 2017).

The outer level includes $J(c,x,x,r(x)) \equiv c(x),$3 and function extensionality, while the inner level carries full homotopy-type-theoretic rules, in particular univalence of each universe (Annenkov et al., 2017). A conversion operation

$J(c,x,x,r(x)) \equiv c(x),$4

embeds inner types and terms into outer ones (Annenkov et al., 2017).

The linking axioms between levels are especially relevant to axiomatic type theory. In minimal 2LTT, the conversion morphism $J(c,x,x,r(x)) \equiv c(x),$5 preserves substitutions and context extension, and preserves $J(c,x,x,r(x)) \equiv c(x),$6, $J(c,x,x,r(x)) \equiv c(x),$7, and $J(c,x,x,r(x)) \equiv c(x),$8 up to canonical isomorphism (Annenkov et al., 2017). In a stronger HTS-style presentation one adds axioms such as injectivity of $J(c,x,x,r(x)) \equiv c(x),$9 on types, strict preservation of $H(c,x): J(c,x,x,r(x)) = c(x),$0, $H(c,x): J(c,x,x,r(x)) = c(x),$1, and $H(c,x): J(c,x,x,r(x)) = c(x),$2, preservation of $H(c,x): J(c,x,x,r(x)) = c(x),$3, $H(c,x): J(c,x,x,r(x)) = c(x),$4, and $H(c,x): J(c,x,x,r(x)) = c(x),$5 up to isomorphism, and the requirement that each outer universe $H(c,x): J(c,x,x,r(x)) = c(x),$6 is fibrant, i.e. in the image of $H(c,x): J(c,x,x,r(x)) = c(x),$7 (Annenkov et al., 2017).

The paper explicitly describes 2LTT as embodying an “axiomatic-type-theory paradigm”: one starts with a core constructive system together with a subsystem of fibrant types and a minimal conversion map, and then organizes additional axioms in layers (Annenkov et al., 2017). This layered architecture is not identical to ATT as defined in 2025, but it is presented as another way of building type theory by controlled addition of axioms rather than by a single monolithic judgemental equality discipline. That connection is explicit in the source material and not merely interpretive (Annenkov et al., 2017).

4. Categorical semantics: path categories and display path categories

A major recent semantic development is the biequivalence between path categories and weak models of axiomatic Martin-Löf type theories (Otten et al., 19 Mar 2025). Path categories are categories equipped with:

• a terminal object,
• distinguished classes of fibrations and weak equivalences,
• pullbacks of fibrations along arbitrary maps,
• sections of trivial fibrations,
• a 2-out-of-6 property for equivalences,
• path-object factorizations of diagonals

$H(c,x): J(c,x,x,r(x)) = c(x),$8

(Otten et al., 19 Mar 2025).

From this structure one obtains mapping-path-object factorizations, a notion of homotopy, coincidence of weak equivalences with homotopy equivalences, and slice-wise path-category structure (Otten et al., 19 Mar 2025). The central result states that the 2-category $H(c,x): J(c,x,x,r(x)) = c(x),$9 is biequivalent to a 2-category of contextual comprehension categories equipped with:

• weakly stable based axiomatic Id-types,
• weakly stable $=$0-types with $=$1- and $=$2-rules,
• suitable structure-preserving morphisms and 2-cells (Otten et al., 19 Mar 2025).

The proof proceeds in both directions. Given a path category, its clan comprehension category yields weakly stable based Id-types via path objects, and $=$3-types via composition of display maps (Otten et al., 19 Mar 2025). Conversely, from a contextual comprehension category with weakly stable Id- and $=$4-structure one extracts homotopy equivalences and verifies the path-category axioms (Otten et al., 19 Mar 2025). The assignments extend to inverse 2-functors up to coherent pseudonatural transformations (Otten et al., 19 Mar 2025).

The same paper introduces display path categories as a refinement in which a subclass of fibrations is singled out as display maps. Such a structure still models based axiomatic Id-types through chosen display path objects, but it does not yet guarantee $=$5-types with $=$6, because composites of display maps need not be display (Otten et al., 19 Mar 2025). To recover fuller type-theoretic structure, one may additionally require homotopy $=$7- and $=$8-types, after which the display path category matches comprehension categories with weakly stable axiomatic Id-, $=$9-, and $\Gamma \vdash A\ \mathrm{type}$0-types (Otten et al., 19 Mar 2025).

This semantic picture is significant because it shows that axiomatic identity types are not merely a syntactic weakening; they correspond to a homotopically motivated categorical structure in which equivalence is primitive and substitution is only specified up to isomorphism before strictification (Otten et al., 19 Mar 2025).

5. Two-dimensional semantics and soundness

A second semantic line uses higher-categorical machinery. The 2025 paper on display-map 2-categories formulates a 2-categorical semantics for dependent type theory with computation axioms (Spadetto, 9 Jul 2025). A display-map 2-category $\Gamma \vdash A\ \mathrm{type}$1 is a $\Gamma \vdash A\ \mathrm{type}$2-category with a chosen 2-terminal object and a class of display maps satisfying three structural requirements:

Requirement	Content
2-pullbacks	Display maps are strictly stable under pullback or pseudo-pullback
Cloven isofibration	Each display map carries chosen lifts of 2-cells
Splitness	These choices respect identities and composition

On top of this structure, the paper adds 2-dimensional data for logical rules. Axiomatic Id-types are represented by a display map over $\Gamma \vdash A\ \mathrm{type}$3 together with an arrow-object 2-cell exhibiting the object of arrows of $\Gamma \vdash A\ \mathrm{type}$4; axiomatic $\Gamma \vdash A\ \mathrm{type}$5-types are represented by a homotopy-equivalence in the slice $\Gamma \vdash A\ \mathrm{type}$6; axiomatic $\Gamma \vdash A\ \mathrm{type}$7-types arise from a relative 2-adjoint to pullback along a display map; and the nullary type formers arise as bireflections (Spadetto, 9 Jul 2025).

The main soundness statement is Theorem 4.26 together with Theorem 2.25: given a display-map 2-category endowed with axiomatic Id, $\Gamma \vdash A\ \mathrm{type}$8, $\Gamma \vdash A\ \mathrm{type}$9, $a,a':A$0, and function-extensionality data, all stable under pullback, its underlying 1-category is a split display-map category that models exactly ATT with propositional $a,a':A$1 axioms (Spadetto, 9 Jul 2025). The proof sketch states that the 2-dimensional structures yield the choice functions for formation, introduction, elimination, and propositional computation, while the cloven isofibration structure turns pseudo-sections into genuine sections (Spadetto, 9 Jul 2025).

This framework is compared explicitly with Garner’s semantics for intensional MLTT. In the intensional case, stronger requirements are imposed: normal isofibrations rather than cloven ones, closure of $a,a':A$2-types under composition up to injective equivalence rather than general homotopy-equivalence, and stronger right biadjoint conditions for $a,a':A$3-types (Spadetto, 9 Jul 2025). The axiomatic semantics is obtained precisely by dropping these normality and injectivity requirements (Spadetto, 9 Jul 2025). This suggests that ATT occupies a strictly weaker but semantically broader position than standard intensional MLTT.

6. Metatheory, models, and non-admissibility results

One of the clearest metatheoretic consequences recorded in the supplied material is that weakening computation to propositional equality preserves semantic generality while blocking recovery of stricter computation principles. The 2025 display-map 2-category paper states that the main technical payoff of axiomatic dependent type theory is that one recovers the usual metatheoretic properties of type checking—“decidability, confluence, …”—while remaining compatible with genuinely intensional models such as groupoids (Spadetto, 9 Jul 2025).

The same paper proves a sharp separation result: the judgemental computation rule for intensional identity types is not admissible in ATT (Spadetto, 9 Jul 2025). The semantic witness is a revisitation of the Hofmann–Streicher groupoid model. In that model:

• objects are small groupoids,
• display maps are Grothendieck fibrations arising from functors or pseudofunctors into $a,a':A$4,
• pullback is given by the Grothendieck construction,
• cloven isofibrations come from usual cleavages,
• the arrow-object for identity is induced by the hom-groupoid $a,a':A$5,
• $a,a':A$6-types and $a,a':A$7-types are interpreted by the indicated categorical constructions (Spadetto, 9 Jul 2025).

All propositional $a,a':A$8 axioms hold in this model, but the corresponding judgemental $a,a':A$9-rules are not forced, because the cleavages are not normal (Spadetto, 9 Jul 2025). In particular, $\Gamma \vdash a =_A a'\ \mathrm{type}$0 and $\Gamma \vdash a =_A a'\ \mathrm{type}$1 remain distinct as 1-cells, with only a non-trivial isomorphism between them (Spadetto, 9 Jul 2025). This yields a semantic proof that strict identity-type computation cannot be derived from the axiomatic system.

A complementary metatheoretic picture appears in the heterogeneous-equality setting. There, the axioms are shown consistent by interpretation in Agda with --with-K, and the presentation inherits normalization and canonicity from the host theory; in particular, closed terms of type $\Gamma \vdash a =_A a'\ \mathrm{type}$2 reduce to numerals (Pitts, 2019). Although this result concerns a specific axiomatization of heterogeneous equality rather than ATT in full generality, it demonstrates that propositional computation can be combined with strong metatheoretic behavior in a concrete implementation (Pitts, 2019).

Two-level type theory contributes a different metatheoretic point. Its conservativity results show that the minimal assumptions of 2LTT are enough to guarantee that no new inner-level theorems are provable beyond those of plain HoTT, while stronger assumptions allow one to internalize semantical constructions such as a universe of semisimplicial types (Annenkov et al., 2017). This exhibits a recurring pattern in axiomatic type-theoretic design: a minimal core preserves conservativity, while additional axioms enlarge the range of internal constructions.

7. General semantic frameworks and research significance

A broader abstract setting for these developments is given by the general framework for the semantics of type theory (Uemura, 2019). There, a type theory is defined as a small representable-map category, i.e. a cartesian category equipped with a stable class of exponentiable arrows closed under identities, composition, and pullback (Uemura, 2019). A model of such a theory is a representable-map functor into the category of discrete fibrations over a small category, with morphisms satisfying a Beck–Chevalley condition (Uemura, 2019).

Within this framework, several general results are established:

• every type theory has a bi-initial model,
• every model has an internal language,
• the category of theories over a type theory is bi-equivalent to a full sub-2-category of models of the type theory (Uemura, 2019).

The bi-initial model is constructed from the Yoneda embedding and the “heart” of its image (Uemura, 2019). The internal language functor sends a model to the set of global sections of each interpreted fibration (Uemura, 2019). The resulting bi-adjunction between theories and models restricts to a bi-equivalence on democratic models, where every object is obtained by iterated context extension from the terminal object (Uemura, 2019).

This abstract semantics does not define axiomatic type theory specifically, but it supplies a unifying language in which Martin-Löf type theory, two-level type theory, and cubical type theory can all be treated within one notion of theory and model (Uemura, 2019). A plausible implication is that axiomatic type theory can be studied not only as a weakening of computation rules, but also as one point in a larger categorical design space of representable-map semantics.

Taken together, the supplied works present axiomatic type theory not as a single formalism but as a coherent research program. In syntax, it replaces definitional computation by proof-relevant computation axioms (Spadetto, 9 Jul 2025, Otten et al., 19 Mar 2025). In semantics, it is modeled by path categories, display path categories, and display-map 2-categories (Otten et al., 19 Mar 2025, Spadetto, 9 Jul 2025). In layered formulations such as two-level type theory, it supports controlled addition of extensionality and fibrancy principles while preserving conservativity under minimal assumptions (Annenkov et al., 2017). In general semantics, it fits into a wider abstract theory of type theories and their internal languages (Uemura, 2019). These results collectively place axiomatic type theory at a technically precise interface between dependent syntax, homotopical semantics, and higher-categorical model theory.